I am in a college algebra course right now and I am having some interesting thoughts about different sizes of infinity. Try to stick with me. My father has thrown in some possible ideas as well and I would like to hear what you think about this. You need to keep an open mind here and think about my father's counter-rationale. If what he says is all true then it makes you think. Here is the situation:

Imagine a coordinate plane, there are infinite points, so imagine infinite "solutions" if you will.

Now imagine a linear line. It also has infinite points/solutions. So you have coordinate infinity and a linear infinity. But not every single infinite coordinate point will satisfy the equation, even though there are infinite solutions to the line.

So this implies that there are smaller and larger values of infinity. You have the "all" infinity of the coordinate plane with a "lesser" infinity being the solutions to the line.

This is where it gets complicated if I can recall everything my father said. My father is thinking that this linear line still contains the coordinate plane infinity becuase he gave the idea that the linear line bends as it moves through space (just as light bends when it goes around planets, or drastically bends around quasars or black holes, etc...) so he thinks that it could be possible that if this "linear" line continuosly bends for infinity then it will eventually fill infinity even in 3D space. So in this sense a linear line will have the coordinate infinity as part of the solution. However, you cannot think in terms of rigid mathematical algebraic thought though.

Couple things, a line ... can only be linear. If it's not linear, then it's not a line, it's a curve. And this only applies in 2D space. If it's in 3D space and still, "A line", that means one of the coordinate value dimensions is 0 and this isn't possible in reality for things made of matter.

Now, in mathematics when we say something is "infinitely long" it doesn't mean it goes on forever, necessarily, it's just for our frame of reference it goes on for as far as we can tell.

This is an important point, because it changes what "infinity" means. In Algebra, infinity is a place holder. It is also in Calculus. You can say something approaches infinity (also know as a limit) But you can't say something IS infinity. It's NOT a value!

Infinity is just a nomenclature to mean end of the spectrum, or forever.

The idea is that, you can't list out every theoretical positive coordinate ... because numbers don't have an end ... I can always add a new decimal place. This is the idea of infinity in mathematics. There's no final number. So, this idea is the only way we can express "the end" or something near it.

NOW, this is not the same as talking about INFINITE in physics or reality 3D Space-time. Entirely different concepts. We have no idea if anything is infinite. We're pretty sure the universe is, in fact, not infinite. There has to be an edge somewhere if it follows physical properties. Because it's still expanding (as far as we can tell) if it was infinite, that wouldn't make any sense.

Just to touch on another "infinite" idea ... alternate realities are said to be "infinite" because ... again ... we have no way of saying, "oh yes, there is actually some limit value that once we reach it there won't be anymore"

Infinite means ... we don't know how far this goes, but we do know ... it goes for as far as we can tell ... so we assume it continues.

So, what I'm saying is, it sounds like either your dad is talking about something different or you guys are using infinite wrong. Again, infinity is not a value. Numbers can only approach it, they can never "be" it. This is why we have more complex mathematics which don't talk about definite values, but trends. Because we can't say what happens once you get to what could be the end of the line.

For sets of objects, even infinite sets, the "standard" idea of determining whether they have the same number of elements is as follows:

Two sets are said to have the same number of elements if it is theoretically possible to make an exact correspondence between them (ie: there exists a mapping between them that is both one-to-one and onto). You can see this makes it possible to say that two infinite sets have the same number of elements even though you can't count them.

It is actually a surprising fact that, under this definition, that there are indeed infinite sets that have different numbers of elements. For example, the infinite set of natural numbers {1,2,3,4,...} has "less" elements than the infinite set of points in a line. The reason for this is Cantor's diagonal argument, which was discovered in the late 1800's and was mindblowing to everyone at the time.

Strangely, the line and the plane have the same number of elements, and one example of a correspondence between them is a space filling curve.

As far as I know, the notion of inequivalent infinite sets is how the phenomenon of zero-point energy is explained. Somehow by placing conductive plates very close together (I am fuzzy on the details here) a "smaller" infinity is created which means a lower ambient radiation pressure between the plates and a force works to push them together. By keeping these two plates apart but in close proximity they hope to harvest usable energy from this phenomenon. I seem to recall reading an article which suggested that specific types of brain cells contain structures which exhibit zero-point attraction.

It's my view that diffraction and gravitational lensing are one in the same, one occurring around individual atoms and the other around massive cosmic bodies, respectively. The view of light as waves is all well and good until they interact with something (photon is "observed,") the superposition potential collapses and then there must be a discrete path from the source, which in both the cases of diffraction and gravitational lensing may bend around solid matter. I would suggest, though, that from the perspective of the photon that path was at no point anything but a perfectly straight line.

The image of the sun in the sky trails the gravitational pull of the sun by the amount of time it takes the light to reach us, about 8.3 minutes or 20 arcseconds (i.e. gravity's effects are instantaneous, not limited by a force-carrier particle like gravitons, as once theorized.) This is a proven fact, and the theory that follows is that, as Einstein presented with general relativity, the effects of gravity are a consequence of matter distorting the fundamental medium, prima mater, or the fabric of space-time, best described as "spooky action at a distance." For objects which aren't changing directions the distortion of the medium is relatively static, but when a massive object moves suddenly it creates ripples that propagate away at some speed limit which is faster than light!

Somewhat off-topic but UFO's supposedly fly by creating a bubble of space which rides on waves of gravitation, the craft sits in the middle of the field and therefore the occupants experience none of the effects of acceleration, enabling maneuvers which would turn the pilot into red jelly otherwise.

Now imagine a linear line. It also has infinite points/solutions. So you have coordinate infinity and a linear infinity. But not every single infinite coordinate point will satisfy the equation, even though there are infinite solutions to the line.

No math background here, but what's keeping you from saying there could be more than one linear infinity? You could have an infinite set of linear infinities that are pointing in all different directions, thus making up a coordinate set and all remaining linear!

You could have an infinite set of linear infinities that are pointing in all different directions, thus making up a coordinate set and all remaining linear!

I suppose you could have infinite linear equations, however I am just referring to any single given linear equation. I was just referring to the solutions of a single linear line. I believe what you said is true though if you allow for more than one equation.

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I don't really know what anyone is saying here

If it is any help to you I'll try to simplify what I started out saying. My idea was just that:

If you have any linear line then there would be infinite points on it. However, even though it has infinite points that does not mean that ANY point will work. May lose you here but: If you have a line y=x, just a line with a positive slop of 1 that passes through the origin a possible solution would be (0,0) but for example the point (0,1) would not be a solution since it is not on the line. So it is a different type of infinity. (Probably doesn't help you much but maybe a little? )

Right, algebra. If you have a line L which lies on a plane P, then the set containing all solutions to L would be a subset of the set of solutions to P. The line is embedded in the plane but it still remains a line. Both are infinite because by definition of a line and a plane, no matter how far you go you can always count one further.

This is a great discussion, infinity is one of those topics people can talk about forever

One really strange thing about space-filling curves and fractals is the fact that the distance between any two points is infinite, because you're taking the limit toward infinity of the function, or in other words you're iterating a function that convolutes the curve, making it longer, an infinite number of times.

I found the name of the conductive plates falling toward eachother, the Casimir effect. It's the result of the wavelength of fluctuations in the vacuum energy of space being restricted to half-integer multiples of the distance between the plates {L, 3/2L, 2L, 5/2L, etc}, making a countable infinity of standing waves in that space, while outside the plates the wavelengths are unrestricted and an uncountable infinity of standing waves with more frequencies exerts more pressure on the plates!

Depends on what step you are on in a corporate ladder. I could tell you horror stories about how upper management views profits and their numerical system viewpoints. Somehow the closer to the top you get, infinite seems more realistically obtainable.

It's easy to do if you divide by zero (which is what you're technically doing if you divide by an unknown value). Undefined values, what's that!?

If you've ever seen a 1 = 2 proof online, they inadvertently divided by zero. It's a sneaky thing too. Because it crops up in places you wouldn't expect. Like dealing with cosine and sine, sometimes people cancel them when in reality the result of a cosine can be zero.

The funny thing is though ... in the physical world ... I'm not sure that 1 DOESN'T equal 2 sometimes. I've seen some crazy quantum stuff. (like objects in 2 places at once ... not at a quantum scale, either) Makes you think maybe you could logically divide by zero and we just don't have a model for the new behavior that happens.

But in straight logic based mathematics ... yeah you divided by zero to make whatever crazy equivalence true. (If I prove 1 is equal to 2, then 2 + 2 is equivalent to 2 + 1, which is 3, and 1 + 1, which is 2. But it's an interesting idea ... because, you could argue in a zero undefined system that 2 + 2 is equal to 2, 3, and 4. The crazy part is this could actually be true!)

Business math is not actual math. It's best case scenario potential positive everything ... used in the context that those are the numbers they should be. It's entirely unethical and vastly damaging behavior in most situations its used in.

I find it difficult to hear an argument that people's lives can be financially or mathematically quantifiable, but it happens every day.

I suppose the worse part is ... it doesn't HAVE to be done like that. They just choose to do it that way because it yields the largest net profit in the shortest amount of time. But you lose significant values in the process. Like refusing to payout someone's life insurance policy even though it was a negligent death.

Business math is not actual math. It's best case scenario potential positive everything ... used in the context that those are the numbers they should be. It's entirely unethical and vastly damaging behavior in most situations its used in.

I find it difficult to hear an argument that people's lives can be financially or mathematically quantifiable, but it happens every day.

I suppose the worse part is ... it doesn't HAVE to be done like that. They just choose to do it that way because it yields the largest net profit in the shortest amount of time. But you lose significant values in the process. Like refusing to payout someone's life insurance policy even though it was a negligent death.

*sighs*So true, so true.People think I drive because I have to. They think I am out here alone because I am too stupid to do anything else but the fact is, I needed to be out here. If I were in management one more day I prolly would have ate a bullet. or worse, made someone else eat one.

I despise driving but when compared to the mechanations of working closely with corporate America.. I just couldn't deal.

That's true working for a government institution too. Our school district is supposed to filled with educated people. But I have never seen so many stupid people in my life.The mismanaged financial numbers are staggering. Over 1/3 of a billion dollars in debt and a shrinking tax base. That's why I hate math....it gets ugly when you deal with that stuff.Makes me pray the next few years go quick so I can leave. AH..corporate America, hmmm...Get up in the morning and try to figure out how screw a few more thousand people out of their retirements, or sell them some other useless piece of BS they don't really need.

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